Impure math

When Samuel Hansen said in his interview “You’re not a pure mathematician” I agreed without thinking, but later the statement bothered me a little. I know what he meant: considering the two categories of pure math and applied math, you’d put yourself in the latter category. Which is true.

But the term “pure” math can be misleading, as if everyone else does impure math. Applied math is not an alternative to theoretical math. Applied mathematicians prove theorems etc. We work on applications in addition to doing what is expected of pure mathematicians. The difference between pure and applied math is motivation, not content. Applied math is motivated by direct application to non-mathematical problems. Pure math seeks to advance math for its own sake. Both are important.

Statistics uses the terms “theoretical” and “applied” rather than “pure” and “applied.” Math doesn’t use “theoretical” as an antithesis to “applied” because applied math is theoretical. But unlike math, being “applied” in statistics does mean you’re often (too often?) excused from proving theorems. The first time I was a coauthor on a statistics paper I was surprised to find out you could publish with just simulation results and no theorems. This happens in applied math as well, but not nearly as often as it does in applied statistics.

On the other hand, when I hear the term “applied statistics” I want to ask “Is there any other kind?” Statistics is applied (and theoretical!) though some statisticians work more directly on applications than others. As Andrew Gelman quips, the difference between theoretical and applied statisticians is that

I assume that statement wasn’t meant to be taken literally, but I agree with the sentiment that the distinction between theoretical and applied statistics can be exaggerated. I’d say the same applies to pure and applied math.

11 thoughts on “Impure math”

I think Marc Kac my have said that all good Mathematics is Applied Mathematics. After a bit of reflection, this appears to be true. Most pure Math of today was motivated by practical problems, and still address abstracted versions of these problems. This is the case for Physics, which has literally sprung into existence areas like Analysis, Topology, Differential Geometry, Probability, and some abstract algebra. Computer Science has been the source for maybe the greatest and most consequential open problem today (P=NP). And then there is Statistics.

The opposite view is Halmos': an applied mathematician is a bad methematician. But if one reads Halmos argument, it had do do with rigor and not subject matter. Many applied mathematicians are fully rigorous in their research.

For these reasons, I don’t believe in the distinction between pure and applied Mathematics. There is good and bad Mathematics; the former being Mathematics that inspires more Mathematics.
There is Mathematics that is directly motivated by applications. But that in itself is not a strong qualifier.

Paul Halmos has an article with the attention-getting title “Applied mathematics is bad mathematics” – you can get a feel for his real opinions by reading this excerpt from an AMS Notices article about him, which quotes him at length: “Not only, as is universally admitted, does the applied need the pure, but, in order to keep from becoming inbred, sterile, meaningless, and dead, the pure needs the revitalization and the contact with reality that only the applied can provide.”

As a lad, wandering through the small topics of undergrad school, I aspired for a brief time to what I imagined to be the lofty heights of purity. Then, I hit ordinary DE with then interim chairman of the math department at GSU, Fred Massey. The problem was that the good doctor described himself as an “applied mathematician” (his non-teaching, consulting “outside” job was in complex, partial DE systems for the rate-flow analysis and control of industrial chemical processes…can we all say “yikes” to that?).

Solving problems for the class, Dr. Massey filled the entire board with a solution based on the methods of the course text and clearly explain it in those terms. Next he made the point we (students) were presumed to have other knowledge derived from pre- or co-requisite courses not obviously directly related to the materials presented in the current textbook – from those principles, he filled half the board with yet another solution both simpler and clearer. Finally, he said that this was the kind of thing he did lots in his role as – you guessed it – an applied mathematician, and he could see a “shortcut.” He filled a quarter board with a third solution, the best and most elegant and still easily within our grasp as students. His breadth and depth of insight and capacity to invoke enlightenment in mere undergrads was peerless. It gave me pause about the use of the word “mere” in conjunction with the expression “applied mathematics.”

After experiencing a couple of courses under Dr. Massey, et al., both applied and pure by personal reports, I graduated and wandered into Actuarial Science for more than a few years in pursuit of ridiculous paychecks. Eventually, I recognized that I didn’t really care as much about topics such as Fermat, still unproven at the time and with which I’d played for quite a while, as I did ARIMA modelling of asset/liability matching. I also accepted that “applied” did not mean “lesser” I lost my conceit that “pure” meant “better.” Certainly there was no less rigor demanded by my highly paid labors in the vineyard of the applied.

Don’t get me wrong, pure math owns great virtues in pushing forward the boundaries of mathematics. However, I now conceive of it as I do pure research in the sciences; if solid, it inflates the sphere of knowledge and eventually ends up being applied in engineering. As with Hardy’s, “Nothing I have ever done is of the slightest practical use,” the validity of all such statements regarding purity passes with time. There reliably come some bright, young men, who recognize a correspondence of the abstract structure of pure mathematics to the actual structure of some “real thing.”

Mainly, I think it’s a debate between the Meyers/Briggs personality sub-types INTP and INTJ with a bit of flip-flopping from time-to-time. Every so, often the pure chap does something practical. Occasionally the applied fellow has to generate a few snippets of new theory to extend a previously incomplete bit of pure work. The rest of the time a -P type sticks to his guns preference-wise and some -J guy finds a practical application.

I shall continue to admire mathematicians, both pure and applied alike, to exactly the extent that their formulations are demonstrably correct.

I prefer the (apparently older) term “speculative mathematics” instead of “pure”. Instead of applications being dirty, non-applied mathematics is simply mathematics that we don’t know how we’ll use yet.

What’s truly annoying about the term “pure mathematics” is that we all know pure mathematicians are being paid by someone to pursue their “art” — usually under the assumption that mathematics is a kind of science, that will eventually have an impact on the economy. To suggest that people who are doing something that matters to other people as “impure” is to be ungrateful for the privilege behind one’s job.

(I.e., if mathematics should be done for its own sake, then why should the taxpayers be paying you?)